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Monochromatic vector scattering by a smooth PEC obstacle: phasor Maxwell fields, the PEC trace condition and outgoing-correction uniqueness

Physical scope

Conditional vector scattering of a smooth frequency- $ω$ source-free vacuum phasor pair by a smooth bounded stationary perfect electric conductor with connected exterior: the scattered and total phasor pairs, the associated real fields solving the zero-source smooth vacuum Maxwell equations on the exterior, the ideal PEC boundary condition $n \times E_{t o t} = 0$ , and uniqueness of the outgoing scattered solution with the prescribed tangential trace. Excluded: lossy, impedance, nonsmooth or dispersive obstacles; determination of induced surface charge or current; radiation-zone amplitude asymptotics beyond the outgoing modal class; singular or point sources; finite-energy, coupled self-force and every empirical claim.

Statement

Assume the Axiom of Choice as carried by Existence and uniqueness for the exterior vector Helmholtz problem with prescribed tangential trace and outgoing radiation, whose variational, Sobolev-compactness and compact-spectral steps use it; the algebraic computations below use no further choice. Let $O \subset R^{3}$ be a bounded open set with $C^{\infty}$ boundary, $D : = R^{3} ∖ \overline{O}$ connected, $k > 0$ , $ω : = c k$ with $c$ the vacuum speed of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and $R > 0$ with $\overline{O} \subset B_{R} (0)$ ; let $V \subseteq R^{3}$ be open with $\overline{D} \subseteq V$ , and let $({\tilde{E}}_{i n c}, {\tilde{B}}_{i n c})$ be a $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pair on $V$ (Frequency- $ω$ source-free vacuum phasor pairs and their real-field encoding, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values). Plane-wave instances are provided by Dispersion, transversality and the two polarization conditions for plane-wave solutions: if $κ \cdot e = 0$ , $b = κ \times e / ω$ and $∣ κ ∣ = ω / c$ , then $({\tilde{E}}_{i n c}, {\tilde{B}}_{i n c}) (x) = (e e^{i κ \cdot x}, b e^{i κ \cdot x})$ satisfies the four phasor equations on $R^{3}$ (Real plane-wave fields encoded by complex amplitudes). Adopt the ideal perfectly conducting (PEC) boundary model: the tangential component of the total electric field vanishes on $\partial O$ . This is a declared idealization of the stationary interface conditions of Stationary macroscopic interface jump conditions for D, B, E and H with vanishing fields inside the conductor; it is not one of the finite-conductivity branches of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch, and the induced surface charge and current are neither prescribed nor determined here. Let $e_{T}$ be the tangential projection of $- {\tilde{E}}_{i n c}$ on $\partial O$ , and let ${\tilde{E}}_{s c a t}$ be the unique $C^{\infty}$ field on $\overline{D}$ of Existence and uniqueness for the exterior vector Helmholtz problem with prescribed tangential trace and outgoing radiation with $(Δ + k^{2}) {\tilde{E}}_{s c a t} = 0$ and $div {\tilde{E}}_{s c a t} = 0$ on $D$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Divergence and curl of a $C^{1}$ vector field), tangential trace $e_{T}$ on $\partial O$ and that lemma's outgoing modal condition on $S_{R}$ . Set ${\tilde{B}}_{s c a t} : = (i ω)^{- 1} curl {\tilde{E}}_{s c a t}$ and, on $D$ , ${\tilde{E}}_{t o t} : = {\tilde{E}}_{i n c} + {\tilde{E}}_{s c a t}$ , ${\tilde{B}}_{t o t} : = {\tilde{B}}_{i n c} + {\tilde{B}}_{s c a t}$ . Then: (a) [Phasor Maxwell] $({\tilde{E}}_{s c a t}, {\tilde{B}}_{s c a t})$ and $({\tilde{E}}_{t o t}, {\tilde{B}}_{t o t})$ are $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pairs on $D$ ; and for every $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pair $(\tilde{E}, \tilde{B})$ on an open set $O^{'} \subseteq R^{3}$ , the associated real fields of Frequency- $ω$ source-free vacuum phasor pairs and their real-field encoding satisfy on $O^{'} \times R$ the smooth-branch vacuum Maxwell equations of The microscopic vacuum Maxwell equations: smooth and weak branches with $ρ \equiv 0$ and $J \equiv 0$ . Consequently the associated real total fields $E_{t o t}, B_{t o t}$ satisfy those source-free equations on $D \times R$ . (b) [PEC boundary condition] the tangential trace of ${\tilde{E}}_{t o t}$ on $\partial O$ vanishes; hence for every unit normal field $n$ of $\partial O$ the associated real total electric field $E_{t o t} (x, t) = Re ({\tilde{E}}_{t o t} (x) e^{- i ω t})$ (Real and imaginary parts, complex conjugation, and modulus) satisfies $n \times E_{t o t} = 0$ on $\partial O \times R$ . (c) [Uniqueness] if $({\tilde{E}}^{'}, {\tilde{B}}^{'})$ is a frequency- $ω$ source-free vacuum phasor pair on $D$ whose fields belong to $C^{\infty} (\overline{D}; C^{3})$ that is outgoing on $S_{R}$ in the modal sense of the vector Helmholtz lemma and has tangential trace $e_{T}$ on $\partial O$ , then $({\tilde{E}}^{'}, {\tilde{B}}^{'}) = ({\tilde{E}}_{s c a t}, {\tilde{B}}_{s c a t})$ on $D$ ; equivalently $({\tilde{E}}_{t o t}, {\tilde{B}}_{t o t})$ is the unique total source-free phasor pair in the corresponding $C^{\infty}$ closure class with that prescribed incident phasor, vanishing tangential electric trace, and outgoing scattered correction ${\tilde{E}}_{t o t} - {\tilde{E}}_{i n c}$ in the lemma's modal class. The total field itself is not required to be outgoing; an incident plane wave is generally not outgoing. (d) [Scope] one positive frequency $ω = c k$ , one smooth bounded stationary PEC obstacle with connected exterior, and $C^{\infty}$ incident and scattered classes only; no lossy, impedance, nonsmooth or dispersive obstacle, no determination of the induced surface charge or current, no radiation-zone amplitude or far-field asymptotics beyond the lemma's outgoing modal class, no point or singular source, no finite-energy or empirical claim.

Facts & Assumptions

Given: The Axiom of Choice; the bounded $C^{\infty}$ obstacle $O$ with connected exterior $D$ ; $k > 0$ , $ω = c k$ , $R > 0$ with $\overline{O} \subset B_{R} (0)$ ; the open $V \supseteq \overline{D}$ and the $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pair $({\tilde{E}}_{i n c}, {\tilde{B}}_{i n c})$ on $V$ ; the tangential projection $e_{T}$ of $- {\tilde{E}}_{i n c}$ on $\partial O$ ; the field ${\tilde{E}}_{s c a t}$ and ${\tilde{B}}_{s c a t} = (i ω)^{- 1} curl {\tilde{E}}_{s c a t}$ of the Statement; the totals ${\tilde{E}}_{t o t} = {\tilde{E}}_{i n c} + {\tilde{E}}_{s c a t}$ and ${\tilde{B}}_{t o t} = {\tilde{B}}_{i n c} + {\tilde{B}}_{s c a t}$ on $D$ .

[F1]

Exterior vector Helmholtz lemma: for the geometry, $k$ and $R$ above and every smooth $C$ -valued tangential field $e_{T}$ on $\partial O$ , there is exactly one $E \in C^{\infty} (\overline{D}; C^{3})$ with $(Δ + k^{2}) E = 0$ on $D$ , $div E = 0$ on $D$ , tangential trace $E_{t a n g e n t i a l} = e_{T}$ on $\partial O$ , and the modal outgoing condition on $S_{R}$ (its trace lies in $H^{1 / 2} (S_{R}; C^{3})$ and its normal derivative on $S_{R}$ equals the sphere Dirichlet-to-Neumann image of that trace componentwise in the weak trace sense); its normal boundary values satisfy the closure relation; the lemma assumes the Axiom of Choice (Existence and uniqueness for the exterior vector Helmholtz problem with prescribed tangential trace and outgoing radiation).

[F2]

Phasor conventions: a frequency- $ω$ source-free vacuum phasor pair on an open set is a $C^{\infty}$ pair $(\tilde{E}, \tilde{B})$ satisfying $div \tilde{E} = 0$ , $div \tilde{B} = 0$ , $curl \tilde{E} = i ω \tilde{B}$ and $curl \tilde{B} = - i ω c^{- 2} \tilde{E}$ there, with divergence and curl acting componentwise on the complex fields; the associated real fields are $E = Re (\tilde{E} e^{- i ω t})$ , $B = Re (\tilde{B} e^{- i ω t})$ (Frequency- $ω$ source-free vacuum phasor pairs and their real-field encoding, Real and imaginary parts, complex conjugation, and modulus).

[F3]

Phasor calculus and equivalence: for $C^{1}$ complex fields the displayed partial-derivative and divergence/curl formulas hold; the real-part evaluation is injective; and the phasor system holds on an open set if and only if the associated real fields satisfy $div E = 0$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl B = c^{- 2} \partial_{t} E$ there (Real parts of $e^{- i ω t}$ phasors: calculus, injectivity and equivalence with the real Helmholtz-Maxwell system).

[F4]

Smooth vacuum Maxwell branch at zero sources: for jointly $C^{1}$ fields with the additional derivatives required by each equation existing and continuous and continuous sources, the adopted vacuum Maxwell postulate consists of the four equations $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ ; with $ρ \equiv 0$ and $J \equiv 0$ this is exactly the source-free system of [F3] (The microscopic vacuum Maxwell equations: smooth and weak branches, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F5]

Plane-wave amplitude equations: for $k \in R^{3} ∖ {0}$ , $ω > 0$ and $e, b \in C^{3}$ , the plane-wave fields satisfy the source-free smooth Maxwell equations if and only if $κ \cdot e = 0$ , $κ \cdot b = 0$ , $κ \times e = ω b$ and $κ \times b = - (ω / c^{2}) e$ ; on the dispersion relation $∣ κ ∣ = ω / c$ the solutions are exactly the pairs with $κ \cdot e = 0$ and $b = κ \times e / ω$ (Dispersion, transversality and the two polarization conditions for plane-wave solutions, Real plane-wave fields encoded by complex amplitudes).

[F6]

Vector-calculus identities: for a $C^{2}$ real field on an open subset of $R^{3}$ , $curl curl F = \nabla div F - Δ F$ and $div (curl F) = 0$ ; divergence and curl are linear over the reals, and both identities and both operators extend to complex fields by applying them to real and imaginary parts componentwise (The curl of a curl is the gradient of the divergence minus the Laplacian, The divergence of the curl of a $C^{2}$ field vanishes, Divergence and curl are linear and satisfy the scalar product rules, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Divergence and curl of a $C^{1}$ vector field).

[F7]

Tangential projection and cross product: for a unit normal field $n$ of $\partial O$ and a field $w$ , the tangential projection is $w_{t a n g} : = w - (w \cdot n) n$ with the Euclidean inner product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , and $n \times w = n \times w_{t a n g}$ because $n \times n = 0$ ; the cross product is bilinear and alternating (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors).

[F8]

Material context: the PEC condition is the declared idealization of the stationary interface conditions of Stationary macroscopic interface jump conditions for D, B, E and H with vanishing fields inside the conductor; the material-closure branches of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch contain no $σ \to \infty$ ideal-conductor branch, and the fields of The electric and magnetic fields as fixed-time spatial vector fields with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values on the laboratory chart of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields are the physical reading adopted by Frequency- $ω$ source-free vacuum phasor pairs and their real-field encoding.

[F9]

Choice accounting: the Axiom of Choice of the Statement is used exactly through [F1]; all other computations are pointwise algebra and finite sums, and no further selection from infinite families is made (The Axiom of Choice).

Proof

technique · direct; instantiate the exterior vector Helmholtz lemma for the scattered field, verify the four phasor equations algebraically, transfer the real-field Maxwell statement through the phasor equivalence, read the PEC condition off the tangential traces, and reduce uniqueness of any candidate to the lemma's uniqueness clause

1.1givenF1F6F7

[Setting and existence of the scattered field] The hypotheses of [F1] are met by the geometry, $k$ and $R$ , and $e_{T}$ is a smooth $C$ -valued tangential field on $\partial O$ by [F7] applied to the $C^{\infty}$ incident field; hence [F1] supplies a unique ${\tilde{E}}_{s c a t} \in C^{\infty} (\overline{D}; C^{3})$ with $(Δ + k^{2}) {\tilde{E}}_{s c a t} = 0$ , $div {\tilde{E}}_{s c a t} = 0$ on $D$ , tangential trace $e_{T}$ on $\partial O$ , and the outgoing modal condition on $S_{R}$ . The field ${\tilde{B}}_{s c a t} : = (i ω)^{- 1} curl {\tilde{E}}_{s c a t}$ is $C^{\infty}$ on $\overline{D}$ by [F6] and the smooth closure regularity of ${\tilde{E}}_{s c a t}$ , and the incident pair together with $e_{T}$ and the geometry is the data of the Statement.

2.1step 1.1F1F2F6algebra

[The scattered pair is source-free] On $D$ : $div {\tilde{E}}_{s c a t} = 0$ by step 1.1; $curl {\tilde{E}}_{s c a t} = i ω {\tilde{B}}_{s c a t}$ by the definition of ${\tilde{B}}_{s c a t}$ ; $div {\tilde{B}}_{s c a t} = (i ω)^{- 1} div (curl {\tilde{E}}_{s c a t}) = 0$ by [F6]; and $curl {\tilde{B}}_{s c a t} = (i ω)^{- 1} curl curl {\tilde{E}}_{s c a t} = (i ω)^{- 1} (\nabla div {\tilde{E}}_{s c a t} - Δ {\tilde{E}}_{s c a t}) = (i ω)^{- 1} k^{2} {\tilde{E}}_{s c a t}$ , which equals $- i ω c^{- 2} {\tilde{E}}_{s c a t}$ because $k = ω / c$ gives $k^{2} / (i ω) = - i k^{2} / ω = - i ω c^{- 2}$ . All four equations of [F2] hold on the open set $D$ for a $C^{\infty}$ pair, so $({\tilde{E}}_{s c a t}, {\tilde{B}}_{s c a t})$ is a frequency- $ω$ source-free vacuum phasor pair on $D$ .

2.2step 1.1F1F2F6algebra

[Uniqueness of the outgoing scattered correction] Let $({\tilde{E}}^{'}, {\tilde{B}}^{'})$ be a frequency- $ω$ source-free vacuum phasor pair on $D$ whose fields belong to $C^{\infty} (\overline{D}; C^{3})$ , outgoing on $S_{R}$ in the modal sense of [F1], with tangential trace $e_{T}$ on $\partial O$ . From the phasor equations and [F6], $curl curl {\tilde{E}}^{'} = i ω curl {\tilde{B}}^{'} = i ω (- i ω c^{- 2} {\tilde{E}}^{'}) = ω^{2} c^{- 2} {\tilde{E}}^{'} = k^{2} {\tilde{E}}^{'}$ and $div {\tilde{E}}^{'} = 0$ , so [F6] gives $\nabla div {\tilde{E}}^{'} - Δ {\tilde{E}}^{'} = - Δ {\tilde{E}}^{'} = k^{2} {\tilde{E}}^{'}$ , that is $(Δ + k^{2}) {\tilde{E}}^{'} = 0$ on $D$ . The field ${\tilde{E}}^{'}$ is $C^{\infty}$ on $\overline{D}$ , has tangential trace $e_{T}$ , divergence zero and the same modal outgoing condition on $S_{R}$ ; all hypotheses of the uniqueness clause of [F1] are therefore satisfied, and ${\tilde{E}}^{'} = {\tilde{E}}_{s c a t}$ on $D$ . Then ${\tilde{B}}^{'} = (i ω)^{- 1} curl {\tilde{E}}^{'} = (i ω)^{- 1} curl {\tilde{E}}_{s c a t} = {\tilde{B}}_{s c a t}$ on $D$ by the third phasor equation and the definition of ${\tilde{B}}_{s c a t}$ .

3.1step 2.1F2F6algebra

[The total pair is source-free] On $D$ the incident pair satisfies the four equations of [F2] because $D \subseteq V$ and the pair is source-free on $V$ ; the scattered pair satisfies them by step 2.1; the totals ${\tilde{E}}_{t o t} = {\tilde{E}}_{i n c} + {\tilde{E}}_{s c a t}$ and ${\tilde{B}}_{t o t} = {\tilde{B}}_{i n c} + {\tilde{B}}_{s c a t}$ are $C^{\infty}$ on $D$ as sums, and each of the four equations is linear over the complex scalars by [F6] applied to real and imaginary parts, so they hold for the totals; hence $({\tilde{E}}_{t o t}, {\tilde{B}}_{t o t})$ is as well a frequency- $ω$ source-free vacuum phasor pair on $D$ .

4.1step 2.1step 3.1F3F4

[Real-field Maxwell statement] Let $(\tilde{E}, \tilde{B})$ be any $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pair on an open $O^{'} \subseteq R^{3}$ . By [F3] its associated real fields satisfy the real system of [F3] on $O^{'} \times R$ , and by [F4] that system with zero sources is exactly the smooth branch of the adopted vacuum Maxwell equations at $ρ \equiv 0$ , $J \equiv 0$ ; the required regularity is the joint $C^{1}$ (indeed $C^{\infty}$ ) calculus supplied by [F3]. Applying this to $({\tilde{E}}_{t o t}, {\tilde{B}}_{t o t})$ on $O^{'} = D$ , which is admissible by step 3.1, gives the asserted source-free equations for $E_{t o t}, B_{t o t}$ on $D \times R$ . This proves the two statements of (a), the scattered and total clauses following from steps 2.1 and 3.1.

4.2step 3.1F2F7algebra

[PEC boundary condition] On $\partial O$ one has ${\tilde{E}}_{t o t, t a n g} = {\tilde{E}}_{i n c, t a n g} + {\tilde{E}}_{s c a t, t a n g} = {\tilde{E}}_{i n c, t a n g} + e_{T} = 0$ , because the tangential projection is the real-linear map of [F7] and $e_{T}$ is the tangential projection of $- {\tilde{E}}_{i n c}$ ; thus the tangential trace of ${\tilde{E}}_{t o t}$ vanishes. For the associated real field, the tangential projection of $E_{t o t} (x, t) = Re ({\tilde{E}}_{t o t} (x) e^{- i ω t})$ is $Re ({\tilde{E}}_{t o t, t a n g} (x) e^{- i ω t}) = 0$ by the real-linearity of the projection [F7] and of $Re$ ; hence $n \times E_{t o t} = n \times E_{t o t, t a n g} = 0$ on $\partial O \times R$ by [F7]. This is (b).

5.1step 3.1step 4.2step 2.2F6algebra

[Equivalent total formulation] Write a candidate total pair as $({\tilde{E}}_{i n c} + {\tilde{E}}^{''}, {\tilde{B}}_{i n c} + {\tilde{B}}^{''})$ on $D$ , with both fields smooth up to its boundary, where $({\tilde{E}}^{''}, {\tilde{B}}^{''})$ is the scattered correction, required to be a $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pair and outgoing on $S_{R}$ in the modal sense, with the total tangential electric trace vanishing. By linearity [F6] the total tangential trace is ${\tilde{E}}_{i n c, t a n g} + {\tilde{E}}_{t a n g}^{''}$ , so the vanishing of the total trace is equivalent to ${\tilde{E}}_{t a n g}^{''} = e_{T}$ ; by step 2.2 applied to $({\tilde{E}}^{''}, {\tilde{B}}^{''})$ , the correction equals $({\tilde{E}}_{s c a t}, {\tilde{B}}_{s c a t})$ on $D$ , hence the total equals $({\tilde{E}}_{t o t}, {\tilde{B}}_{t o t})$ ; conversely the constructed total has these properties by steps 3.1, 4.2 and the outgoing property of ${\tilde{E}}_{s c a t}$ from step 1.1. This is the equivalent formulation of (c).

6.1step 4.2step 2.2step 5.1F5F8F9given∎

[Scope and choice accounting] Clause (d) records the exact hypotheses: one positive frequency, one smooth bounded stationary PEC obstacle with connected exterior, and $C^{\infty}$ incident and scattered classes; no lossy, impedance, nonsmooth or dispersive obstacle, no induced surface charge or current, no radiation-zone asymptotics beyond the modal class, no singular source and no empirical claim enters the argument. The PEC condition is adopted as an idealization by [F8] and is not derived from the material closures; the total field is not asserted to be outgoing, as an incident plane wave generally is not, and the uniqueness statement is exactly about the total-minus-incident correction. The plane-wave instance clause of the Statement is the specialization of the equivalence of [F5] to $∣ κ ∣ = ω / c$ and $b = κ \times e / ω$ , so it introduces no commitment beyond that cited statement. The only choice principle used is the Axiom of Choice assumed in the Statement through [F1], with the accounting of [F9]; all remaining steps are pointwise algebraic identities and finite sums.

Remarks

The phrase "outgoing" applies to the scattered correction only. The constructed total field ${\tilde{E}}_{t o t} = {\tilde{E}}_{i n c} + {\tilde{E}}_{s c a t}$ is source-free on $D$ but need not be outgoing, and no decay or far-field asymptotic beyond the lemma's modal class is claimed.
The ideal PEC condition is a boundary model, not a theorem about any particular conductor; the induced surface charge and current that would accompany it in a microscopic model are neither prescribed nor derived here.

Monochromatic vector scattering by a smooth PEC obstacle: phasor Maxwell fields, the PEC trace condition and outgoing-correction uniqueness

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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